Orders in non-semisimple algebras

In several branches of representation theory, the existence of Auslander-Reiten sequences has led to new structural insights, for example, in the module theory of artinian algebras [11, 6], in the theory of lattices over classical orders [18, 2] over a complete discrete valuation domain R. and for the corresponding derived categories [12, 17]. For an R-order Λ in a finite dimensional algebra A over the quotient field K of R, Auslander and Reiten [2, 5] have characterized the non-projective indecomposable Λ-lattices E for which an Auslander-Reiten sequence (AR-sequence for short) L (rightwards arrow with hook) II ↠ E exists as those Λ-lattices E for which the A-module K E is projective. In the present paper, we shall introduce a modified version of AR-sequences in the category Λ-lat of Λ-lattices which behave similar to AR-sequences of modules over artinian algebras. In fact, there will be a close relationship to AR-sequences in Λ̄-mod, where Λ̄ := Λ/(Rad R)Λ. This relationship extends to AR-sequences in A-mod if Λ is hereditary (e. g. for a path order Λ = RΔ of a quiver Δ without oriented cycles.) Our investigation is inspired by recent work of W. Crawley-Boevey [9] who determined the lattices E with ExtRΔ(E, E) = 0 over a path order RΔ.